EGARCH-Based Sectoral Volatility Forecasting

Seasonality Analysis

Before fitting any model, recurring calendar patterns need to be ruled out. Month plots show whether any calendar month consistently delivers higher or lower returns. Spectral plots decompose the return series by frequency to check for periodic cycles. Friedman tests confirm whether any apparent pattern is statistically significant. If seasonality is absent, non-seasonal ARIMA is sufficient and no seasonal differencing is needed.

Month Plots: All Five Sectors

Monthly log returns by calendar month. No month consistently outperforms or underperforms: seasonal ARIMA is not needed.

Seasonality

Sample Spectrum: All Five Sectors

Periodogram for each sector. Flat spectra with no dominant spike confirm no seasonal cycle in any sector's return series.

Seasonality

Stationarity & Correlation

ARIMA and GARCH models require stationary inputs. ADF, Phillips-Perron, and KPSS tests confirm that monthly log returns are stationary in mean for all five sectors, so no differencing is needed (d=0). ACF and PACF plots then reveal the lag structure of the return series, directly informing ARIMA order selection by showing how much autocorrelation remains at each lag.

ACF: All Five Sectors

Autocorrelation at each lag. Nearly all lags fall within confidence bands: monthly returns have minimal serial dependence, justifying low-order ARIMA.

Stationarity & Correlation

PACF: All Five Sectors

Partial autocorrelation isolates direct lag effects. Mostly insignificant values confirm AR orders above 1 are not needed for any sector.

Stationarity & Correlation

Mean Model Diagnostics: ARIMA & Exponential Smoothing

BIC-selected ARIMA models and exponential smoothing methods (SES and Holt) are fitted to monthly log returns. The goal is white-noise residuals: if all linear structure has been captured in the mean, the remaining variance clustering can be passed to GARCH. Residual time series, ACF plots, and histograms jointly confirm whether each mean model is adequate before volatility modeling begins.

ARIMA Residuals: Time Series

ARIMA residuals over time. Crisis-period spikes show heteroscedasticity that ARIMA cannot model, directly motivating GARCH.

Mean Models

ACF of ARIMA Residuals

All residual ACF lags fall within confidence bands. White noise in the mean is confirmed: GARCH on squared residuals is appropriate.

Mean Models

ARIMA Residual Histograms

Residuals show heavier tails than normal. NIFTY IT has a visible outlier. Leptokurtosis justifies Student-t errors in EGARCH.

Mean Models

SES Residuals: All Sectors

SES residuals mirror ARIMA residuals. Crisis spikes confirm variance clustering is in the data, not the model choice.

Mean Models

Holt Residuals: All Sectors

Holt adds a trend term but residuals are near-identical to SES. All three mean model classes produce white-noise residuals, clearing the path for GARCH.

Mean Models

Volatility Modeling & Crisis Analysis

EGARCH(1,1) is fitted to ARIMA residuals for each sector. EGARCH is preferred over symmetric GARCH because it allows negative shocks to drive more volatility than positive shocks of the same size, the leverage effect well-documented in equity markets. Four crisis periods are then overlaid on the conditional volatility series: GFC 2008, Taper Tantrum 2013, Demonetization 2016, and COVID-19 2020, to quantify how each sector responds under systemic stress.

Rolling 12-Month Volatility: All Sectors

Cross-sector volatility clustering is clearly visible. NIFTY IT peaks highest; GFC and COVID dominate as the two largest broad-market spikes.

Volatility & Crisis

Crisis Volatility: NIFTY BANK

BANK shows the largest GFC spike across all sectors. COVID and GFC both produce sharp surges: banking is the most crisis-sensitive sector.

Volatility & Crisis

Crisis Volatility: NIFTY IT

NIFTY IT has an extreme early-period spike near 1.0 from data history. During GFC and COVID, IT volatility is lower than BANK, reflecting sector resilience.

Volatility & Crisis

Crisis Volatility: NIFTY PHARMA

Pharma spikes sharply during COVID, unlike IT. Defensive sector demand uncertainty during the pandemic drove elevated volatility rather than suppressing it.

Volatility & Crisis

Crisis Volatility: NIFTY AUTO

AUTO is highly cyclical: GFC and COVID both produce sharp spikes. Credit and consumer sentiment transmission makes it more crisis-sensitive than Pharma or FMCG.

Volatility & Crisis

Crisis Volatility: NIFTY FMCG

FMCG has the smallest crisis spikes overall. Demonetization 2016 is notable: FMCG distribution channels were directly disrupted by the cash withdrawal policy.

Volatility & Crisis

EGARCH Volatility Forecasting

Rolling one-step-ahead EGARCH forecasts are generated for each day in the 365-day out-of-sample window using an expanding estimation window, so the model only uses information available at each forecast date. Forecasted conditional volatility is compared against 20-day realized volatility (annualized). RMSE, MAE, and MAPE quantify accuracy across sectors, revealing which sectors have more predictable volatility dynamics.

EGARCH Forecast vs Realized: NIFTY BANK

Forecast tracks direction well but smooths spikes. High persistence (beta1 ~ 0.97) means conditional variance adjusts gradually, not abruptly.

Forecasting

EGARCH Forecast vs Realized: All Five Sectors

Defensive sectors (FMCG, Pharma) show tighter forecast alignment. Cyclical sectors (Bank, Auto) show wider gaps at volatility spikes: persistence explains cross-sector accuracy differences.

Forecasting